Version 1.0.0.1 is available through MediaPortal Extension Manager, it also requires a patch, available in this post.
LogoManager.config file goes in C:\ProgramData\Team MediaPortal\MediaPortal
LogoManager.dll goes in C:\Program Files (x86)\Team MediaPortal\MediaPortal\plugins\Windows

I'm running MP 1.19. I dont find the LogoManager in the Extension Manager.
Under the "Installed Extension"-tab I have LogoManager which I installed from the MP Plugin site, version 1.0.0.0.
Is it because I', not running the newest MP version?

Perhaps I should switch to MP2 and give it a try (again)
Can I have both MP1 and MP2 running side-by-side, just to test MP2 without removing/overwritting anything in MP1?

Nope, LogoManager will work fine on old MP as I'm on 1.3 myself. It's because noone is using installation from site nowadays snce we have Extensions Manager as app and Extensions plugin in MediaPortal itself. I didn't test the case with upgrading from MP site plugin installation but you've tried it seems and it's not working. Last thing you could check is your list of extensions is up-to-date by clicking Upgrade list in Extension Manager's dropdown menu. If you stil can't upgrade I suggest you to delete site's plugin and install LogoManager plugin from Extensions Manager. It should be here.